Surface flow phenomena, such as rain water flowing down a tree trunk and progressive water front in a shower room, are common in real life. However, compared with the 3D spatial fluid flow, these surface flow problems have been much less studied in the graphics community. To tackle this research gap, we present an efficient, robust and high-fidelity simulation approach based on the shallow-water equations. Specifically, the standard shallow-water flow model is extended to general triangle meshes with a feature-based bottom friction model, and a series of coherent mathematical formulations are derived to represent the full range of physical effects that are important for real-world surface flow phenomena. In addition, by achieving compatibility with existing 3D fluid simulators and by supporting physically realistic interactions with multiple fluids and solid surfaces, the new model is flexible and readily extensible for coupled phenomena. A wide range of simulation examples are presented to demonstrate the performance of the new approach.

We use techniques from both real and complex algebraic geometry to study$K$-theoretic and related invariants of the algebra$C$($X$) of continuous complex-valued functions on a compact Hausdorff topological space$X$. For example, we prove a parameterized version of a theorem by Joseph Gubeladze; we show that if$M$is a countable, abelian, cancellative, torsion-free, semi-normal monoid, and$X$is contractible, then every finitely generated projective module over$C$($X$)[$M$] is free. The particular case $$ M = \mathbb{N}_0^n $$ gives a parameterized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic$K$-theory of$C$($X$) follows from the particular case $$ M = {\mathbb{Z}^n} $$ . We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on$C$^{*}-algebras, and for a homology theory of commutative algebras to vanish on$C$^{*}-algebras. These criteria have numerous applications. For example, the vanishing criterion applied to nil$K$-theory implies that commutative$C$^{*}-algebras are$K$-regular. As another application, we show that the familiar formulas of Hochschild–Kostant–Rosenberg and Loday–Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of$C$($X$). Applications to the conjectures of Beĭlinson-Soulé and Farrell–Jones are also given.

We compute a momentum space version of the entanglement spectrum and entanglement entropy of general Young tableau states, and one-point functions on Young tableau states. These physical quantities are used to measure the topology of the dual spacetime geometries in the context of gauge/gravity correspondence. The idea that Young tableau states can be obtained by superposing coherent states is explicitly verified. In this quantum superposition, a topologically distinct geometry is produced by superposing states dual to geometries with a trivial topology. Furthermore we have a refined bound for the overlap between coherent states and the rectangular Young tableau state, by using the techniques of symmetric groups and representations. This bound is exponentially suppressed by the total edge length of the Young tableau. It is also found that the norm squared of the overlaps is bounded above by inverse powers of the exponential of the entanglement entropies. We also compute the overlaps between Young tableau states and other states including squeezed states and multi-mode entangled states which have similarities with those appeared in quantum information theory.

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